That said, thousands of sites run on Windows Server without any issues. A poorly-configured server with any OS will be vulnerable; Linux won't save you from a poor admin.
So I guess my "best practice" for deplying an ASP.NET app would be to use Windows Server 2008 (likely Web edition). And hire a good administrator.
You can use Mono to run ASP.NET applications on Apache/Linux, however it has a limited subset of what you can do under Windows. As for "they" saying Windows is more vulnerable to attack - it's not true. IIS has had less security problems over the last couple of years that Apache, but in either case it's all down to the administration of the boxes - both OSes can be easily secured. These days the attack points are not the OS or web server software, but the applications themselves.
It depends what specific .NET technologies you're using. The Mono Project provides an Apache module (mod_mono) for running ASP.NET sites, and from what I gather it works well.
Mono doesn't support all the .NET APIs, though - notably WPF (and possibly WCF too, I can't remember) - but it does provide good support for much else of the framework.
If you're starting from scratch and particularly want to target non-Windows servers, then ensuring your project works with Mono would be a good goal to aim for. However, if you need particular APIs or language features that are not supported by Mono, then you will need to use a Windows server for deployment. It's a design-time/architectural choice that should make up front.
Now a days .Net is run in multiple platforms,like linux ,Mac os etc.
but mono is not fully platform independent ,Because to deploy .NET in another OS required
third party software.so it is not like java platform independent.
Mono is running in different platform ,because of JIT is there in different os.
Mono is not fully success in moonlight(silver light in .NET) .Not only Research is going on.
Mono uses XSP2 server or apache . some of the big companies are using this project,Some of the robotic project are also running on mono.
The Mono project is your best option. However, it has a lot of pitfalls (like incomplete API support in some areas), and it's legally gray (people like Richard Stallman have derided the use of Mono because of the possibility of Microsoft coming down on Mono by using its patent rights, but that's another story).
Anyway, Apache supports .NET/Mono through a module, but the last time I checked the version supplied with Debian, it gave Perl language support only; I can't say if it's changed since, perhaps someone else can correct me there.
I can speak from experience. Even if your ASP.net website only uses .NET libraries supported by Mono you are going to have a hard time getting it to run if its anything beyond Hello World.
You won't have to re-write much code but you will spend hours/days/weeks dealing with little issues with mod_mono/xsp/apache configuration and file permissions and error handling and all the little things that go into a large website. (Be prepared to spend a lot of time asking questions on serverfault :) )
The problem is that a lot of people don't use Mono for ASP.net websites and so there aren't as many people reporting bugs so a lot of things that are minor bugs go un-fixed for a long time.
So I know this is an older question but I think it could stand an updated answer.
Microsoft has officially released asp.net vnext and its open source and deploy-able to both Linux and Mac. Its all still pretty new but does rely on the latest builds of mono and thus currently needs you to compile the mono-framework but in time I suspect that it will be easier to access as various linux distros release updated versions of mono. This is a how to setup guide
This information may be somewhat volatile and with updates is due to change.
dotnet is the official home of .NET on GitHub. It's a great starting point to find many .NET OSS projects from Microsoft and the community, including many that are part of the .NET Foundation.